Kinematic analysis of five-link spherical mechanisms

Mechanism and Machine Theory, 1974, Vol. 9, pp. 181-190. Pergamon Press. Printed in Great Britain

Kinematic Analysis of Five-Link Spherical Mechanisms D. S. Tavkhelidze* and N. S. Davitashvili Received 20 March 1973 Abstract The kinematic analysis of spherical five-link mechanisms is shown as a general solution. As special cases, solutions are obtained for the planar five-link mechanism, the planar four-link mechanism, and the spherical four-link mechanism. Introduction IN THE design of modern machines and apparatus and in other branches of industry spatial, spherical and planar mechanisms are widely applied. Five-link jointed mechanisms with two degrees of mobility are one kind of this class of mechanisms. During recent years the problems of analysis of five-link jointed mechanisms have attracted the attention of many scientists. This is caused by the fact that, in contrast to the four-link mechanisms, five-link mechanisms may accomplish considerably greater varieties of motion transformation and curve tracing. Special cases of the problem of spatial, spherical and planar five-link mechanisms have been considered in the works of Allen[7], Dobrovolski[2], Kirkhof[8], Levitski[3], Tavkhelidze[5], Duffy and Habib-Olahi[9], Ovakimov [4], Davitashvili[6] and others. The present work is devoted to the kinematic analysis of spherical and planar fivelink mechanisms with two degrees of mobility.

Analysis A spherical five-link jointed mechanism is shown in Fig. 1. The lengths 11, 12, 13, 14and Is of links 1, 2, 3, 4, and 5 of the mechanism are expressed by spherical distances. The following condition must be observed for the lengths of the spherical mechanism's links: 0 < Ii < 7r where l~ is the length of the i-th link. The angles of rotation, ~2 and q~, of the cranks 2 and 5 will be given for the kinematic study of spherical five-link mechanisms, in addition to the lengths of the links. The position of link 3 relative to link 2 is determined by the expression Kl

q~3= arccos ~ + arccos

K3 K:K----~'

(1)

*Dr. tech. Nauk Professor, Chair of Theory of Machines and Mechanisms, Georgian Polytechnical Institute by V. I. Lenin, Tbilisi, 79 Carlov Street 109 app 20, USSR.

181

182

Figure 1. where K~ = m~ + m2 c o s ~5 + m3 c o s ~o2 - m4 c o s ~02 C O S q~s - - m5 s i n ~02 s i n ~o5; K2 = %/m6 -

rtl7 c o s 2 tp2 -

m s COS 2 ~ s - - m 9 c o s 2 ~02 c o s 2 ~os + ~ 1 0 COS ~02

- - m l l COS ~ 5 - - m 1 2 COS q)2 c o s

~05 - - m l 3 s i n ~02 s i n q~5 ~t_ m 1 4 c o s 2 q)2 c o s q)5

+ m~5 s i n 3~02 s i n q~5; K3 =/31t9 + m20 cos ~2-

m21 c o s ~ 5 - m 2 2 c o s ~01 c o s ~05 -

K 4 = s i n 13. Further m, = sin/2cos

l, c o s 15;

m2 = s i n l~ s i n

12 s i n

15;

m3 = s i n l~ c o s b_ c o s 15; m4 = s i n 15 c o s l, c o s 12; m5 = s i n h c o s 12; m 6 = 1 - - COS 2 Ii COS 2 12 COS 2 15 - - s i n 2 12 Sill 2 15;

m7 = s i n 2 12(sin 2 1~ c o s 2 1, - s i n 2 15); m s = s i n 2 / 5 ( s i n 2 l, c o s 2 12 - s i n : 12); m9 =

s i n 2 12 s i n 2 15(1 + c o s 2 l~);

m~o = ½ s i n 21~ s i n 21: c o s 2 15; m~l = ~ s i n 21, s i n 21~ s i n 215 c o s 2 12; m ~2 = ~ s i n 212 s i n 215 c o s 2 L ; m,3 = 21s i n 212 s i n 2 h c o s / l ;

m23 s i n ~02 s i n ~ 5 ;

183 ~'~14 "~

½s i n 211 S i l l 2 12 s i n 2 h ;

m15

s i n It s i n 2 12 s i n 2/5;

n'l16

½sin 211 s i n 2/2

~ 1 7 ~-~ s i n

mls

sin 2 1,;

Ii s i n 2/2 sin 2 15;

---- 2 s i n 2 12 s i n 2 15 COS ll;

n'~l 9 ---~ COS 14 -- COS 11 COS 12 COS ~ 2 0 ~--~ s i n

11 s i n 12 c o s 13 c o s 15;

m 2 1 ---~ s i n

l~ s i n 15 c o s 12 c o s / 3 ;

13 c o s 15;

sin l~ s i n 15 c o s l~ c o s / 3 ; m23

sin 12 s i n 15 c o s / 3 .

T h e p o s i t i o n o f l i n k 4 r e l a t i v e t o l i n k 5 is d e t e r m i n e d b y t h e e x p r e s s i o n K5 ~o, = a r c c o s ~ + a r c c o s

K~

K2K-'--~7"

(2)

where K5 = n, - n2 c o s ~2 - n3 c o s ~0, - n , c o s ~2 c o s q~5 - n~ s i n ,~2 s i n ~5; K6 = n6 + n7 c o s ¢p2 - n , c o s q~5 - n9 c o s ~2 c o s ~5 - n~o s i n q~2 s i n q~5; K~ = s i n 1,. Further n~ = s i n 1, c o s 1~ c o s 12; n : = s i n ll s i n 12 s i n l,; n3 = s i n I~ c o s c o s 12 c o s 15; n 4 = s i n 12 c o s It c o s 15;

n5 = s i n 12 c o s 15; n6 = c o s 13 - c o s l, c o s 12 c o s 14 c o s / 5 ; n7 = s i n l~ s i n 12 c o s 14 c o s 15; ns = s i n ll s i n 15 c o s 12 COS 14; n9 = s i n 12 s i n 15 c o s l~ c o s 14; n ~0 = s i n 12 s i n 15 c o s 14. The angular velocities of links 3 and 4 are given by

=/32oJ2;

(3)

0)4 ~ i420)2~

(4)

w3

184

where i3~ and

the ratios of the angular velocities involved, and are given by

i42 a r e

_ lfK~K2-

K,K~

K ; K 2 - K~K& \

i3~ =

K2\ ~

• z42-

I_]_(.K'sK2- K s K ; K ~ - K , K ~ ~. K 2 \ X/-~-~----~5 ~ q N/K~2K72-K6 ~]'

(5)

~-~----~K~K,~---~'~3~);

(6)

to2 is the given angular velocity of the driving link 2, and K;, K~, K;, K~ and K~ are derivatives of the magnitudes K,, K2, K3, Ks and K6 with reference to the generalized coordinate ¢~. The angular accelerations of links 3 and 4 2°!

e3 = to2 132+ i32e2;

(7)

e,, = toz2i~z + i42e2,

(8)

where ih and i~2 are the ratios of the angular accelerations, equal to the derivatives with respect to the same generalized coordinate of the ratios of the above angular velocities. The given angular acceleration of the driving link 2 is represented by e2. With the aid of certain angular velocities and accelerations the linear velocities and accelerations of specific points of the mechanism are easily determined. If the spherical distances, expressed in the obtained formulae, are expressed by the spherical radius O and the corresponding chords L,, then the above expressions will assume the following form. Expression (1) K,

K3

q~3= arc cos ~ + arc cos K2K"--~4'

(9)

where K~ = M~ +

M2 cos

K2 =

- M 7 c o s 2 q~2 -

~/M6

q~5+ M3 cos q~2-

M4

COS q~2COS ~Os-- Ms sin q~2sin Cs;

M8 cos 2 q~5-

M 9 c o s 2 ~ 2 c o s 2 q~5 + M l o c o s q~2

- M . cos q~5- Ml2 COS (~2 COS (~5- - M~3 sin (~2 sin q~ + M~4 cos 2 q~2cos q~5 + M~5 sin -

2q~2

sin ~

-

M16 COS (~2 C O S

2 (~5 --

M17

sin q~2sin 2q~5

M~. sin 2q~: sin 2q~;

K3 = Mr9 +

/ K 4 = L3"~[ I ¥

M2o COS q~z-- M21 cos , s - M22 cos q~2cos tp~ L32

402.

Further

MI=L~(I_L12\/

~-~O)( 1

_Ls2\

I- L22.

~-O-~p2)q1 40 2'

M23

sin ,2 sin q~;

185

I[, L,=W, t:=k[, S..,=~

L,L=L~

M3=LI(1

L~=\/ L~'~ [--

Ll2"

-~)U-~)X/~ 4~,

(

L~'~E L5~.

Ll~\l

M4 = L~ l-~p-"~p~}tl-~'~p~]~i/, 4p2, M~ = L , ( I - L==\ l

L~ ~

~'-~p~)~i/1 -4p~;

M6 ~- L , ~ + L : ~ + L~ ~ - L'2(L'~ + 4L2~ + 4L~:) + L~(L:~ + 8Ls~) + L~'

4p: + L,~[L~ 4 + L~" + L,~(L~ ~ + L~:)] + L2:L~:(4L, ~ + 2L~ ~ + 2L~ ~)

4p 4

L,4(L~ 4 + L;') + L ~ L , : [ 2 L ~ L ~ ~ + L,~(4L, ~ + 4 L : ~ + 4L~)]

16p 6 + Lj2L22L52(L12L22 + LI2L52 + L22L52)

Lj4L24L54

16p s L 2

2

L 2

Ls 2

64p 1° ,

M-~ = .~--(1- 4~--.~)[L,~'(1 - ~---~p ~./L':\/L~2~ ~) ~ 1 - ~--~p " _ L~"(1 - 4--~p Ls:~l" ~./1 ' 2

_L,~'~[ 1

L22~2_L22(1

~--~0-~)[ ~(' 4,,,, -~, L d L ~ ~/

L, ~

L, "\/

Ld'~l

M~= ,2 t a - ~ + ~ ) t l - ~ ) t , - 4 , , v ,

L2~'~]

-~,,;

L~'~.

~,o--~L,L~(, -~;c' L,~,, _L4" ,~,~ L: ~ ~;c'-~)~/(1-~)(,-~)~ ml!

=

2LiL5(1

Mn = 2L2L~(1

Ll2\[

- L22 2

L22\1

L52

L52

L22

L12

Ls2

L52

L,:~2 2 L,~]] X [(1- ~fi-~p2]- ~(1 - ~--sp~ij; .Mi3 = 2L2Ls(1

L l2\ l L~\ I L ~ ~/('-~) L~2 ('-~) L~ ~ -~;c'-~;c'~) )( ~')7( ~')(~,)

186

M,7-

2LIL2L~ 2/

L~ 2 + L2 2 + M19

L2~\[

L 2

2

2

2 Ll L2 ~-~ ~1 - ~~p~]~1 - 4--~# 2) ~/(l-~fi~p ) (1-4--~p) ;

--

L3 2 -

L4 2 4-

L5 2

2

L ,2(L22 4- L32 4- L52 ) + L22(L32 + L52) - L32 L5 2

40 2 4-

L , 2 L a 2 ( L 3 2 4- L52) + L32L52(LI 2 4- L22)

804

M2o = t , t 2

(

L~2',/

1 -~p-~p2)~l

/( L,)(

L;~ ~-Gd 1-a-7/

M2, = L,L,(1-~)(1-~) M=~

Expression

L 2 L ~ ( I - L12\[

L12L22L32L52 1606 '

i(1- L':](1 - L52]" 40 ] \ 4o~J' L32

L22

L52

(2) will a s s u m e t h e f o l l o w i n g f o r m

K5 K~ ~#4 = a r c c o s K 2 4- a r c c o s K:K----~/

(10)

where K~ = N , - N2 c o s ¢2 - N3 c o s ¢5

-

N4 c o s ~2 c o s tp5 - - N5 sin ¢= sin ¢5;

K6 = N6 + N7 c o s ~¢2 - N8 c o s ~¢5 - N~ c o s ~¢2 c o s ¢5 - N,0 sin ~¢2 sin ~¢5;

K7 = L 4 x / 1

L4 2 4p 2"

Nl=Ls(1

Ll2

Further "

- ~p-~p 2J ~/1-4p 2;

187 L,L2L5 p2

N2=

N3=L,(1

,

2

1-

N5 = L~(1 -

5

1-

L22\[

L52~ F, L, 2

L'~\[

L,=~ ~ - ~ 2 .

L,2\ /

L22

- ~ ) t , 1- ~ ) V

L~(1

N4

1-

;

'-4o ~;

)V,-402;

l[

N6=~ (L2+L2:-L32+L42+L2) L,2(L22 + L42 + L, 2) + L22(L~ 2 + L, 2) + L42L, 2 20 2 + L12L22(L42 + L'2) + L42L'2(L'2 + L22) - L12L22L42L'2];6

4p 4

N~ = LiL2(I N.

:

N~

8p

L~2\ /

J

L~ 2

-

-3~o)t'-3-Yo];

LiL,(1

L2

L,_Ls(I

LI2\[

-L';~.,/{1

L42

(

g'2X[

g'2'~

L22

Ls:

402J •

Angular velocities o)~ and ~o4and angular accelerations e3 and e4 of links 3 and 4 are determined by corresponding formulae (3), (4), (7) and (8), where the ratios of angular velocities i32 and i~= have the form _

i~2=

I_(K;K2- K,K; K ~ K 2 - K~K& ~. K2k ~ + X/K22K~-K3=] ' I [K'sK2- K~K"

K~K2 Z K~K'~ X/Kz2K:~-K62] '

i,n:-~-]2 \ ~3-2~_~--~5~ ~-

(!l) (12)

here K~, K;, K~, K~ and K~ are derivatives of magnitudes K~, K2, K3, K5 and K6 with respect to the generalized coordinate. As a special case, the expressions for the spherical mechanism may be reduced to those for planar five-bar mechanisms (Fig. 2). If the spherical radius p-*o% one obtains a five-link plane mechanism and the expression for its kinematic analysis.

188

4

L3

B

L5

Lz

Figure 2.

T h e p o s i t i o n of link 3 of the p l a n a r five-link m e c h a n i s m A B C D E (Fig. 2) will b e d e t e r m i n e d for e x p r e s s i o n (9), w h i c h for the g i v e n special case is as f o l l o w s Ki

¢3 = arc cos ~

K3

+ arc cos -----K'Ks ,

(13)

where K~ = M~ + M3 cos Cs - M4 cos (~2 - ~5); K2 = X/M6+M~o cos ¢ 2 - M , , cos ¢ 5 - M , 2 cos ( ¢ s - q ~ ) ; K3 = M ~ 9 + Mso cos ¢2 - Ms, cos ¢5 - M22 cos (¢2 - q~5); K4 = L3.

Further Mj = L:;

M2 = M 7 = M8 = M9 = M14 . . . . . M3 = L~; M 4 - - M5 = L~;

M6 = LI 2 + L22 + L~2;

M,o = 2L~L,_; Mll = 2LIL5; M12 = Ml3 =

2L2L~;

M~9 = ½(L,2 + L22+ L32 - L , 2 + L52);

M2o = L~Ls; M2~ = L~L~; M22

= M23

=

L2Ls.

M18 = 0;

189

The position of link 4 of the planar five-link mechanism will be determined from expression (10), which for the given special case will look as follows K5

K6

¢4 = arc cos ~ + arc cos K2K----~7'

(14)

where K5 = Nz - N3 cos q~5- N4 cos (q~2- q~5); K 6 -- N 6 + N 7 c o s q~2 - N 8 c o s ff~5-- N 9 c o s (q92 -- q05);

K 7 ~--- L 4 . Further N~ = L5; N~=0; N3 = L,; N4 = N5 = L2; N6 = '(L,~ + L 2 2 - L32+ L42+ L52); N7 = LIL2; N8 = L,Ls; N g = N , o = L2Ls. Angular velocities to3 and 094 and angular accelerations E3 and E, of links 3 and 4 of the planar five-link mechanism are determined according to the formulae (3), (4), (7) and (8), where the ratios i32 and i42 of the angular velocities are determined with the aid of expressions (1 1) and (12).

Conclusion From the c o m m o n expressions, mentioned for the solution of the problem of a five-link spherical mechanism, with the exception of the special case described above, the solution of a four-link spherical and four-link plane mechanisms' motion can also be obtained. Block diagrams and programs have been composed for the reduced expressions, and by this means the problems of analysis of five-link spherical and planar mechanisms have been solved on the electronic computer BESM-4. Thus, the problem of the kinematic analysis of spherical and planar five-link mechanisms with two degrees of freedom has been solved. The given solution of the problem of a spherical five-link mechanisms' kinematic analysis is presented as a general program (as special cases there can be obtained: the solution of the problem of the analysis of a planar five-link mechanism, a planar fourlink mechanism and a spherical four-link mechanism). These expressions serve as a basis for the further study of the synthesis and dynamics of the mentioned mechanisms.

190

References [1] A R T O B O L E V S K I I. I., Theory of Mechanisms, M. (1965). [2] D O B R O V O L S K I V. V., Trajectories of five-link m e c h a n i s m , Works of Machine-Tool Institute, Vol. II. M o s c o w (1938). [3] L E V I T S K I N. I., A p p r x o m a t e synthesis of jointed m e c h a n i s m s with two degrees of freedom, Works of the Machine-Building Institute, M, 21, 83-84 (1961). [4] O V A K I M O V A. G., The problem of space m e c h a n i s m s ' positions with several degrees of f r e e d o m and its solution by m e t h o d s of clased vector profile. Mekh. Mashinost., M, 29-30 (1971). [5] T A V K H E L I D Z E D. S., T h e study of plane five-link m e c h a n i s m s with two degrees of mobility, Metsniereba, Tbilisi (1972). [6] D A V I T A S H V I L I N. S., On the problem of kinematic study and designing of five-link jointed m e c h a n isms, Paper, Tbilisi (1972). [7] A L L E N G. W., T h e design of linkages to generate function of two variables. Trans. ASME, Ser. B, 81, J. Engng Ind. (1959). [8] K I R C H H O F M., E n t w u r f Ffinfgliedrigen Zwiekurbelgetriebe unter Beriicksichtigung des Ubertragungswinkels, Marchinenbautechnik 12, 6 (1963). [9] D U F F Y J. and H A B I B - O L A H I I--I. I., A displacement analysis of spatial five-link 3R-2C m e c h a n i s m - - I . On the closures of the R C R C R m e c h a n i s m s . J. Mechanisms 6 (3) (1971).

K H H E M A T H q E C K H I T I AHA.YlH3 I I f l T H 3 B E H H b l X C @ E P H q E C I ( H X

MEXAHH3MOB

~ . C. TABXEYlFI~3E n H. C. ~ A B H T A I I I B H . r l H P e 3 m M e - B CTaTbe np14BO/114TC~I petueHne 3a~a~la I~riHeMaTrlqecKoro aHanvi3a cqbeprl,iecKHx 14 nsiocK14x I'I~ITH3BeHHblX mapn14pHbix MexaHri3MoB c ~ByMlff cTenen~iMri I'IO~BPl)KHOCTI4artannxmfecKrtM MeTO~IOM. ~ n a onpe~lenen14n nono~rermn 3BenbeB 3 r~ 4 (p14c. 1) cqbepH,tecroro nnTrt3Bemmro m a p r m p n o r o Mex&H143Ma nony,~em,i Bbtpa~eHnn (1) n (2), KOTOpbIe B~,me~tenb~ c rtp14MeHe14rleM cqbepa,JecKo~ Tp14ronoMeTpn14. Onpellene14bi TaK~re yrYlOBble cKopocTH 14 ycKopeHH~ 3BeUbeB 3 I4 4. Bbxpa~eHr~n, onpe~tennK~tttne nono~re14ne 3BeHbeB 3 14 4, Bblpa~KeHbl TaK~e n p n rioMomi,i pa,a14yca cqbep~,[ ¢